The displacement of a body has two components: a rigid-body displacement and a deformation.

A rigid-body displacement consists of a simultaneous translation and rotation of the body without changing its shape or size.

Deformation implies the change in shape and/or size of the body from an initial or undeformed configuration to a current or deformed configuration (Figure 1).

A change in the configuration of a continuum body can be described by a displacement field. A displacement field is a vector field of all displacement vectors for all particles in the body, which relates the deformed configuration with the undeformed configuration. Relative displacement between particles occurs if and only if deformation has occurred. If displacement occurs without deformation, then it is deemed a rigid-body displacement.

The displacement of particles indexed by variable i may be expressed as follows. The vector joining the positions of a particle in the undeformed configuration and deformed configuration is called the displacement vector. Using in place of and in place of , both of which are vectors from the origin of the coordinate system to each respective point, we have the Lagrangian description of the displacement vector:

Where is the unit vector that defines the basis of the material (body-frame) coordinate system.

Expressed in terms of the material coordinates, the displacement field is:

The partial derivative of the displacement vector with respect to the material coordinates yields the material displacement gradient tensor. Thus we have,

The material deformation gradient tensor is a second-order tensor that represents the gradient of the mapping function or functional relation , which describes the motion of a continuum. The material deformation gradient tensor characterizes the local deformation at a material point with position vector , i.e. deformation at neighbouring points, by transforming (linear transformation) a material line element emanating from that point from the reference configuration to the current or deformed configuration, assuming continuity in the mapping function , i.e. differentiable function of and time , which implies that cracks and voids do not open or close during the deformation. Thus we have,

Consider a particle or material point with position vector in the undeformed configuration (Figure 2). After a displacement of the body, the new position of the particle indicated by in the new configuration is given by the vector position . The coordinate systems for the undeformed and deformed configuration can be superimposed for convenience.

Consider now a material point neighboring , with position vector . In the deformed configuration this particle has a new position given by the position vector . Assuming that the line segments and joining the particles and in both the undeformed and deformed configuration, respectively, to be very small, then we can express them as and . Thus from Figure 2 we have

where is the relative displacement vector, which represents the relative displacement of with respect to in the deformed configuration.

For an infinitesimal element , and assuming continuity on the displacement field, it is possible to use a Taylor series expansion around point , neglecting higher-order terms, to approximate the components of the relative displacement vector for the neighboring particle as

Calculations that involve the time-dependent deformation of a body often require a time derivative of the deformation gradient to be calculated. A geometrically consistent definition of such a derivative requires an excursion into differential geometry[2] but we avoid those issues in this article.

The time derivative of is

where is the velocity. The derivative on the right hand side represents a material velocity gradient. It is common to convert that into a spatial gradient, i.e.,

where is the spatial velocity gradient. If the spatial velocity gradient is constant, the above equation can be solved exactly to give

assuming at . There are several methods of computing the exponential above.

Related quantities often used in continuum mechanics are the rate of deformation tensor and the spin tensor defined, respectively, as:

The rate of deformation tensor gives the rate of stretching of line elements while the spin tensor indicates the rate of rotation or vorticity of the motion.

To transform quantities that are defined with respect to areas in a deformed configuration to those relative to areas in a reference configuration, and vice versa, we use Nanson's relation, expressed as

where is an area of a region in the deformed configuration, is the same area in the reference configuration, and is the outward normal to the area element in the current configuration while is the outward normal in the reference configuration, is the deformation gradient, and .

The corresponding formula for the transformation of the volume element is

Derivation of Nanson's relation

To see how this formula is derived, we start with the oriented area elements

Figure 3. Representation of the polar decomposition of the deformation gradient

The deformation gradient , like any second-order tensor, can be decomposed, using the polar decomposition theorem, into a product of two second-order tensors (Truesdell and Noll, 1965): an orthogonal tensor and a positive definite symmetric tensor, i.e.

where the tensor is a proper orthogonal tensor, i.e. and , representing a rotation; the tensor is the right stretch tensor; and the left stretch tensor. The terms right and left means that they are to the right and left of the rotation tensor , respectively. and are both positive definite, i.e. and , and symmetric tensors, i.e. and , of second order.

This decomposition implies that the deformation of a line element in the undeformed configuration onto in the deformed configuration, i.e. , may be obtained either by first stretching the element by , i.e. , followed by a rotation , i.e. ; or equivalently, by applying a rigid rotation first, i.e. , followed later by a stretching , i.e. (See Figure 3).

It can be shown that,

so that and have the same eigenvalues or principal stretches, but different eigenvectors or principal directions and , respectively. The principal directions are related by

Several rotation-independent deformation tensors are used in mechanics. In solid mechanics, the most popular of these are the right and left Cauchy-Green deformation tensors.

Since a pure rotation should not induce any stresses in a deformable body, it is often convenient to use rotation-independent measures of deformation in continuum mechanics. As a rotation followed by its inverse rotation leads to no change () we can exclude the rotation by multiplying by its transpose.

The IUPAC recommends[4] that the inverse of the right Cauchy-Green deformation tensor (called the Cauchy tensor in that document), i. e., , be called the Finger tensor. However, that nomenclature is not universally accepted in applied mechanics.

Earlier in 1828,[7]Augustin Louis Cauchy introduced a deformation tensor defined as the inverse of the left Cauchy-Green deformation tensor, . This tensor has also been called the Piola tensor[4] and the Finger tensor[8] in the rheology and fluid dynamics literature.

Therefore the uniqueness of the spectral decomposition also implies that . The left stretch () is also called the spatial stretch tensor while the right stretch () is called the material stretch tensor.

The effect of acting on is to stretch the vector by and to rotate it to the new orientation , i.e.,

In a similar vein,

Examples

Uniaxial extension of an incompressible material

This is the case where a specimen is stretched in 1-direction with a stretch ratio of . If the volume remains constant, the contraction in the other two directions is such that or . Then:

Derivatives of the stretch with respect to the right Cauchy-Green deformation tensor are used to derive the stress-strain relations of many solids, particularly hyperelastic materials. These derivatives are

Let be a Cartesian coordinate system defined on the undeformed body and let be another system defined on the deformed body. Let a curve in the undeformed body be parametrized using . Its image in the deformed body is .

The concept of strain is used to evaluate how much a given displacement differs locally from a rigid body displacement.[1][9] One of such strains for large deformations is the Lagrangian finite strain tensor, also called the Green-Lagrangian strain tensor or Green – St-Venant strain tensor, defined as

or as a function of the displacement gradient tensor

or

The Green-Lagrangian strain tensor is a measure of how much differs from .

The Eulerian-Almansi finite strain tensor, referenced to the deformed configuration, i.e. Eulerian description, is defined as

or as a function of the displacement gradients we have

Derivation of the Lagrangian and Eulerian finite strain tensors

A measure of deformation is the difference between the squares of the differential line element , in the undeformed configuration, and , in the deformed configuration (Figure 2). Deformation has occurred if the difference is non zero, otherwise a rigid-body displacement has occurred. Thus we have,

In the Lagrangian description, using the material coordinates as the frame of reference, the linear transformation between the differential lines is

Then we have,

where are the components of the right Cauchy-Green deformation tensor, . Then, replacing this equation into the first equation we have,

or

where , are the components of a second-order tensor called the Green – St-Venant strain tensor or the Lagrangian finite strain tensor,

In the Eulerian description, using the spatial coordinates as the frame of reference, the linear transformation between the differential lines is

where are the components of the spatial deformation gradient tensor, . Thus we have

where the second order tensor is called Cauchy's deformation tensor, . Then we have,

or

where , are the components of a second-order tensor called the Eulerian-Almansi finite strain tensor,

Both the Lagrangian and Eulerian finite strain tensors can be conveniently expressed in terms of the displacement gradient tensor. For the Lagrangian strain tensor, first we differentiate the displacement vector with respect to the material coordinates to obtain the material displacement gradient tensor,

Replacing this equation into the expression for the Lagrangian finite strain tensor we have

or

Similarly, the Eulerian-Almansi finite strain tensor can be expressed as

The stretch ratio is a measure of the extensional or normal strain of a differential line element, which can be defined at either the undeformed configuration or the deformed configuration.

The stretch ratio for the differential element (Figure) in the direction of the unit vector at the material point , in the undeformed configuration, is defined as

where is the deformed magnitude of the differential element .

Similarly, the stretch ratio for the differential element (Figure), in the direction of the unit vector at the material point , in the deformed configuration, is defined as

The normal strain in any direction can be expressed as a function of the stretch ratio,

This equation implies that the normal strain is zero, i.e. no deformation, when the stretch is equal to unity. Some materials, such as elastometers can sustain stretch ratios of 3 or 4 before they fail, whereas traditional engineering materials, such as concrete or steel, fail at much lower stretch ratios, perhaps of the order of 1.001 (reference?)

The diagonal components of the Lagrangian finite strain tensor are related to the normal strain, e.g.

where is the normal strain or engineering strain in the direction .

The off-diagonal components of the Lagrangian finite strain tensor are related to shear strain, e.g.

where is the change in the angle between two line elements that were originally perpendicular with directions and , respectively.

Under certain circumstances, i.e. small displacements and small displacement rates, the components of the Lagrangian finite strain tensor may be approximated by the components of the infinitesimal strain tensor

Derivation of the physical interpretation of the Lagrangian and Eulerian finite strain tensors

The stretch ratio for the differential element (Figure) in the direction of the unit vector at the material point , in the undeformed configuration, is defined as

where is the deformed magnitude of the differential element .

Similarly, the stretch ratio for the differential element (Figure), in the direction of the unit vector at the material point , in the deformed configuration, is defined as

The square of the stretch ratio is defined as

Knowing that

we have

where and are unit vectors.

The normal strain or engineering strain in any direction can be expressed as a function of the stretch ratio,

Thus, the normal strain in the direction at the material point may be expressed in terms of the stretch ratio as

solving for we have

The shear strain, or change in angle between two line elements and initially perpendicular, and oriented in the principal directions and , respectivelly, can also be expressed as a function of the stretch ratio. From the dot product between the deformed lines and we have

where is the angle between the lines and in the deformed configuration. Defining as the shear strain or reduction in the angle between two line elements that were originally perpendicular, we have

A representation of deformation tensors in curvilinear coordinates is useful for many problems in continuum mechanics such as nonlinear shell theories and large plastic deformations. Let be a given deformation where the space is characterized by the coordinates . The tangent vector to the coordinate curve at is given by

The three tangent vectors at form a basis. These vectors are related the reciprocal basis vectors by

Let us define a second-order tensor field (also called the metric tensor) with components

The problem of compatibility in continuum mechanics involves the determination of allowable single-valued continuous fields on bodies. These allowable conditions leave the body without unphysical gaps or overlaps after a deformation. Most such conditions apply to simply-connected bodies. Additional conditions are required for the internal boundaries of multiply connected bodies.

The necessary and sufficient conditions for the existence of a compatible field over a simply connected body are

We can show these are the mixed components of the Riemann-Christoffel curvature tensor. Therefore the necessary conditions for -compatibility are that the Riemann-Christoffel curvature of the deformation is zero.

No general sufficiency conditions are known for the left Cauchy-Green deformation tensor in three-dimensions. Compatibility conditions for two-dimensional fields have been found by Janet Blume.[16][17]